Quantum cascades in nano-engineered superconductors: geometrical, thermal and paramagnetic effects

The effect of a parallel magnetic field on the orbital motion of electrons in high-quality superconducting nanowires resulting in a superconductor-to-normal transition which occurs through a cascade of jumps in the order parameter as a function of the magnetic field. Such cascades originate from the transverse size quantization that splits the conduction band into a series of subbands. Here, based on a numerical solution of the Bogoliubov-de Gennes equations for a hollow nanocylinder, we investigate how the quantum-size cascades depend on the confining geometry, i.e., by changing the cylinder radius R and its thickness d we cover the range from the nanowire-like to the nanofilm-like regime. The cascades are shown to become much less pronounced when increasing R/d, i.e., when the nanofilm-like regime is approached. When the temperature is non-zero they are thermally smoothed. This includes the spin-magnetic-field interaction which reduces the critical (depairing) parallel magnetic field H(c,//) but does not have any qualitative effect on the quantum cascades. From our calculations it is seen that the paramagnetic limiting field H(par) significantly exceeds H(c,//) even in extremely narrow nanocylinders, i.e., when R,d are down to a few nanometers, and H(c,//) is only about 10% larger when switching-off the spin-magnetic-field interaction in this case. Both characteristic fields, H(c,//) and H(par), exhibit pronounced quantum-size oscillations. We demonstrate that the quantum cascades and the quantum-size oscillations survive in the presence of surface roughness.